In this paper, we propose a novel approach to establish the convergence of the time-domain perfectly matched layer (PML) method for acoustic scattering problems. The time discretization employs the second-order backward differentiation formula (BDF2) combined with convolution quadrature (CQ), while the spatial domain is truncated by a circular PML layer. We establish a rigorous error estimate that accounts for both the temporal discretization error and the PML truncation error. The temporal error converges at the optimal rate O(τ ^2) under suitable regularity assumptions on the data. The PML error is shown to decay exponentially in σ _0 d/τ , where σ _0 is the absorption parameter, d is the layer thickness, and τ is the time step. A distinctive feature of the analysis is that setting the absorption coefficient proportional to 1/τ makes the PML attenuation automatically strengthen as the time step is refined, so that the exponential decay dominates any algebraic growth factors.
When the 3D axisymmetric Stokes problem is reduced to a 2D problem with a Neumann boundary condition, we design a family of divergence-free, conforming finite element method on rectangular meshes. In this method, the 2D velocity is approximated by continuous Q_k+2,k× Q_k,k+1 polynomials and the pressure is approximated by discontinuous Q_k+1,k polynomials on rectangular meshes. This approach produces divergence-free solutions for the velocity. Stability and quasi-optimal convergence are proved for this family of finite elements, and numerical tests confirm both the theory and the pressure-robustness of the method.
Numerical solution of parabolic PDEs with nonsmooth initial conditions often exhibits reduced and/or unstable convergence orders due to the so-called quantization error. For a class of nonsmooth initial conditions, including representative payoffs of financial derivatives, we study how high-order convolution-based smoothings modify the quantization error. For each initial condition and smoothing operator, we derive closed-form expressions for the leading order terms of the quantization error. Examining the dependence of the quantization error leading coefficient on the relative location of the discontinuity with respect to the discretization grid, we identify the minimum smoothing order required to guarantee stable fourth-order convergence (with or without grid alignment), when using fourth-order discretizations in space and time. These results explain when extrapolation-based acceleration methods, including Richardson extrapolation and the sparse grid combination approach, succeed or fail.
We propose a two-stage, structure-preserving numerical scheme for the nonlinear Schrödinger–Poisson system. The first stage combines a high-order Gauss collocation method in time with a conforming finite element method in space, achieving arbitrarily high-order accuracy. The second stage applies a post-processing correction at each time level to enforce discrete mass and energy conservation. An induction argument establishes local unique solvability of the post-processing system, discrete mass and energy conservation, and the optimal-order error estimate O(τ ^k+1+h^p) in the L^∞ (0,T;H^1) norm, where k and p denote the polynomial degrees of the temporal and spatial finite element approximations, respectively. Numerical experiments validate the theoretical results and demonstrate the accuracy and robustness of the proposed method for long-time simulations. In particular, a cosmological sine-wave collapse benchmark with singular time-dependent coefficients demonstrates the performance of the scheme in a demanding collapse regime.
This paper is concerned with the first application of the augmented Lagrange method for the reconstruction of the Robin coefficient in a parabolic equation. We first transform the inverse problem into a constrained minimization problem and establish the existence of minimizers. Then, we propose a novel unconstrained saddle-point problem based on an augmented Lagrangian formulation, and derive its equivalence to the constrained minimization problem. The saddle-point problem is discretized using the finite element method in space and Euler backward difference method in time, and the convergence of the fully discretized problem is rigorously proved. Finally, we apply an Uzawa algorithm to solve the discrete saddle-point problem and present several numerical examples to illustrate its accuracy and efficiency.
